What Is Multiplication? Equal Groups and Arrays

Counting equal groups

$$5 \times 7 = 35$$

That means five groups of seven. Five bags with seven marbles in each. Count them all and you get $35$.

Adding would work too:

$$7 + 7 + 7 + 7 + 7 = 35$$

Multiplication is a shorter way of writing that. It only works because the groups are equal — five bags holding different amounts could not be counted this way.

Arrays show it

An array arranges objects in equal rows.

$$\begin{array}{ccccccc} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \end{array}$$

Five rows of seven dots. There are $35$ dots, so $5 \times 7 = 35$.

The two numbers being multiplied are the factors, and the answer is the product.

Why the order does not matter

Turn that array a quarter turn. It becomes seven rows of five.

Not a single dot was added or taken away, so the total is unchanged:

$$5 \times 7 = 7 \times 5 = 35$$

This is the commutative property, and it is worth more than it first looks.

It halves the times tables. Learning $7 \times 8 = 56$ gives you $8 \times 7 = 56$ for free. Every product you learn comes with a second one attached, so the number of facts to memorize is roughly half what the grid suggests.

Two more properties that help

Splitting a factor. A hard product becomes two easier ones:

$$7 \times 8 = 7 \times (4 + 4) = 28 + 28 = 56$$ $$6 \times 9 = 6 \times (10 - 1) = 60 - 6 = 54$$

That second one is worth keeping. Every nine times is a ten times minus one group, which turns the hardest column into the easiest.

Grouping three factors. When multiplying three numbers, do whichever pair is easier first:

$$2 \times 7 \times 5 = 7 \times (2 \times 5) = 7 \times 10 = 70$$

Working left to right would mean $14 \times 5$. Reordering makes it a ten.

Multiplying by a multiple of ten

$$9 \times 80$$

Split the $80$ into $8 \times 10$:

$$9 \times 8 = 72 \quad\Rightarrow\quad 72 \times 10 = 720$$

Multiply by the single digit, then add the zero. The zero appears because multiplying by ten shifts every digit one place left.

Product	Working	Answer
$5 \times 60$	$5 \times 6 = 30$	$300$
$7 \times 40$	$7 \times 4 = 28$	$280$
$9 \times 80$	$9 \times 8 = 72$	$720$

Word problems with equal groups

The phrase to look for is a rate — each, every, per.

A box holds $6$ pencils. How many in $4$ boxes?

Four groups of six:

$$4 \times 6 = 24$$

Chairs are set out in $8$ rows of $5$. How many chairs?

An array of eight rows:

$$8 \times 5 = 40$$

The word each is the signal that the groups are equal, which is what makes multiplication the right operation.

Worked examples

Common mistakes

Practice problems